Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-2 Chapter Goals After completing this chapter, you should be able to:  Compute the mean and standard deviation for a discrete probability distribution  Use the binomial, hypergeometric and Poisson discrete probability distributions to find probabilities  Describe when to apply the binomial, hypergeometric and Poisson distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-3 Introduction to Probability Distributions  Random Variable  Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-4 Discrete Random Variables  Can only assume a countable number of values Examples:  Roll a die twice Let X be the number of times 4 comes up (then X could be 0, 1, or 2 times)  Toss a coin 5 times. Let X be the number of heads (then X = 0, 1, 2, 3, 4, or 5)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-5 Experiment: Toss 2 Coins. Let X = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H HH Probability Distribution X X Value Probability 0 1/4 = /4 = /4 = Probability

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-6 Discrete Random Variable Summary Measures  Expected Value of a discrete distribution (Weighted Average)  Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = (0 x.25) + (1 x.50) + (2 x.25) = 1.0 X P(X)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-7  Variance of a discrete random variable  Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X X i = the i th outcome of X P(X i ) = Probability of the i th occurrence of X Discrete Random Variable Summary Measures (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-8 Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y.2 Recession- $ 25 - $200.5 Stable Economy Expanding Economy Investment E(X) = μ X = (-25)(.2) +(50)(.5) + (100)(.3) = 50 E(Y) = μ Y = (-200)(.2) +(60)(.5) + (350)(.3) = 95

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-9 Computing the Standard Deviation for Investment Returns P(X i Y i ) Economic condition Passive Fund X Aggressive Fund Y.2 Recession- $ 25 - $200.5 Stable Economy Expanding Economy Investment

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-10 Interpreting the Results for Investment Returns  The aggressive fund has a higher expected return, but much more risk μ Y = 95 > μ X = 50 but σ Y = > σ X = 43.30

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-11 Probability Distributions Continuous Probability Distributions Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions Normal Uniform Exponential Ch. 5Ch. 6

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-12 The Binomial Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-13 Binomial Probability Distribution  A fixed number of observations, n  e.g.: 15 tosses of a coin; ten light bulbs taken from a shipment  Two mutually exclusive and collectively exhaustive categories  e.g.: head or tail in each toss of a coin; defective or not defective light bulb  Generally called “success” and “failure”  Probability of success is p, probability of failure is 1 – p  Constant probability for each observation  e.g.: Probability of getting a tail is the same each time we toss the coin

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-14 Binomial Probability Distribution (continued)  Observations are independent  The outcome of one observation does not affect the outcome of the other  Two sampling methods  Infinite population without replacement  Finite population with replacement

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-15 Examples  A manufacturing plant labels items as either defective or acceptable  A firm bidding for contracts will either get a contract or not  A marketing research firm receives survey responses of “yes I will buy” or “no I will not buy”  New job applicants either accept the offer or reject it

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-16 Rule of Combinations  The number of combinations of selecting X objects out of n objects is where: n! =n(n - 1)(n - 2)... (2)(1) X! = X(X - 1)(X - 2)... (2)(1) 0! = 1 (by definition)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-17 P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2,..., n) n = sample size (number of trials or observations) p = probability of “success” P(X) n X ! nX p(1-p) X n X ! ()!    Example: Flip a coin four times, let x = # heads: n = 4 p = p = (1 -.5) =.5 X = 0, 1, 2, 3, 4 Binomial Distribution Formula

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-18 Example: Calculating a Binomial Probability What is the probability of one success in four flips if the probability of success is.5? X = 1, n = 4, and p =.5

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-19 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Distribution  The shape of the binomial distribution depends on the values of p and n  Here, n = 5 and p =.1  Here, n = 5 and p =.5 (all distributions for p=.5 are symmetrical)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-20 Binomial Distribution Characteristics  Mean  Variance and Standard Deviation Wheren = sample size p = probability of success (1 – p) = probability of failure

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-21 n = 5 p = 0.1 n = 5 p = 0.5 Mean X P(X) X P(X) 0 Binomial Characteristics Examples

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-22 Using PHStat  Select PHStat / Probability & Prob. Distributions / Binomial…

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-23 Using PHStat  Enter desired values in dialog box Here:n = 10 p =.35 Output for X = 0 to X = 10 will be generated by PHStat Optional check boxes for additional output (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-24 P(X = 3 | n = 10, p =.35) =.2522 PHStat Output P(X > 5 | n = 10, p =.35) =.0949

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-25 The Hypergeometric Distribution Binomial Poisson Probability Distributions Discrete Probability Distributions Hypergeometric

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-26 The Hypergeometric Distribution  “n” trials in a sample taken from a finite population of size N  Sample taken without replacement  Trials are dependent  Concerned with finding the probability of “X” successes in the sample where there are “A” successes in the population

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-27 Hypergeometric Distribution Formula Where N = Population size A = number of successes in the population N – A = number of failures in the population n = sample size X = number of successes in the sample n – X = number of failures in the sample (Two possible outcomes per trial)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-28 Properties of the Hypergeometric Distribution  The mean of the hypergeometric distribution is  The standard deviation is Where is called the “Finite Population Correction Factor” from sampling without replacement from a finite population

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-29 Using the Hypergeometric Distribution ■Example: 3 different computers are checked from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded? N = 10n = 3 A = 4 X = 2 The probability that 2 of the 3 selected computers will have illegal software loaded is.30

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-30 Hypergeometric Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Hypergeometric …

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-31 Hypergeometric Distribution in PHStat  Complete dialog box entries and get output … N = 10 n = 3 A = 4 X = 2 P(X = 2) = 0.3 (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-32 The Poisson Distribution Binomial Hypergeometric Poisson Probability Distributions Discrete Probability Distributions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-33 The Poisson Distribution  Apply the Poisson Distribution when:  You wish to count the number of times an event occurs in a given interval  The probability that an event occurs in the interval is the same for all intervals of equal size  The number of events that occur in one interval is independent of the number of events that occur in the other intervals  The average number of events per interval or unit is (lambda)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-34 Poisson Distribution Formula where: X = number of successes per unit = expected number of successes per interval e = base of the natural logarithm system ( )

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-35 Poisson Distribution Characteristics  Mean  Variance and Standard Deviation where = expected number of successes per unit

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-36 Graph of Poisson Probabilities X = P(X = 2) =.0758 Graphically: =.50

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-37 Poisson Distribution Shape  The shape of the Poisson Distribution depends on the parameter : = 0.50 = 3.00

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-38 Poisson Distribution in PHStat  Select: PHStat / Probability & Prob. Distributions / Poisson…

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-39 Poisson Distribution in PHStat  Complete dialog box entries and get output … P(X = 2) = (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-40 Chapter Summary  Addressed the probability of a discrete random variable  Discussed the Binomial distribution  Discussed the Poisson distribution  Discussed the Hypergeometric distribution